Suppose that n is an integer Show that the following three s

Suppose that n is an integer. Show that the following three statements are equivalent

3n + 4 is even.

n + 5 is odd.

n2 is even.

by showing (i) (ii) and (ii) (iii).

Solution

To show that all the 3 statements are eqivalent we need to prove or disprove all three statements simultaneously,

let n=even,

this means n=2k;

thus,

1) 3n+4=3x2k+4=2x3k+4=even => true

2) n+5= 2k+5= 2k+4+1 = even +1= odd =>true

3)n² = (2k)² =even=>true

=> it is equivalent

taking n=odd

this means n=2k+1

thus,

1) 3n+4=3x(2k+1)+4=2x3k+4+3= odd => false

2) n+5= 2k+1+5= 2k+6 = even =>false

3)n² = (2k+1)² = 4k² +4k+1 = odd=>false

all the statements are false

=> it is equivalent